If the pair \((f,\mu )\)  admits an antiderivative, then the set of antiderivatives of \((f,\mu )\)  is an affine space whose underlying vector space is the relative homology group \(\mathrm {H}_1(\Gamma , \Gamma ^d).\)
To prove this result one applies Proposition \(3.12\)  in [1]} to the graph \(\Gamma ^s,\)  with the set of boundary vertices defined as those vertices that belong to \(\Gamma ^d.\)
Auxiliary classification result
In this subsection we follow [1]}. Let \((M,\omega )\)  be a symplectic surface with boundary \(\partial M.\)  Denote by \(CB(M)\subset C^\infty (M)\)  the space of Morse functions on \(M\)  constant on the boundary \(\partial M,\)  and without critical points on the boundary \(\partial M.\)  Elements of \(CB(M)\)  are called functions of \(CB-\) type.
Definition 4.5 A coset \([\alpha ]\in \Omega ^1(M)/\mathrm {d} \Omega ^0(M)\)  is said to be of
\(CB\) -type
if
\(\mathfrak {curl}[\alpha ] \in CB(M).\) 
A coadjoint orbit \(\pazocal O\)  called to be of
\(CB\) -type
if some (and hence every) coset \([\alpha ]\in \pazocal O\)  is of \(CB\) -type.

All definitions from the present paper such as Reeb graph, compatibility conditions, circulation graph, etc. can be modified for the case of functions and cosets of \(CB-\) type, see details in [1]}. The result we are interested in can be formulated as follows.
Theorem 4.6 ([1]}) 

Let \(M\)  be a connected symplectic surface with or without boundary. Then coadjoint orbits of \(\mathrm {SDiff}(M)\)  of \(CB-\) type are in one-to-one correspondence with (isomorphism classes of) circulation graphs
\((\Gamma , f, \mu ,  \pazocal C)\)  compatible with \(M\) .
In other words, the following statements hold:
For a symplectic surface \(M\)  and cosets of \(CB-\) type \([\alpha ], [\beta ] \in \mathfrak {s}\mathfrak {vect}^*(M)\) 
the following conditions are equivalent:

\([\alpha ]\)  and \( [\beta ]\)  lie in the same orbit of the \(\mathrm {SDiff}(M)\)  coadjoint action;

circulation graphs \(\Gamma _{[\alpha ]}\)  and \(\Gamma _{[\beta ]}\)  corresponding to the cosets \([\alpha ]\)  and \([\beta ]\) 
are isomorphic.

For each circulation graph \(\Gamma \)  which is compatible
with \(M\) , there exists a generic \([\alpha ] \in \mathfrak {s}\mathfrak {vect}^*(M)\)  such that \(\Gamma _{[\alpha ]} =(\Gamma , f, \mu ,  \pazocal C).\)
Augmented circulation graph
In the case of surfaces with boundary circulation functions do not form a complete set of invariants for coadjoint orbits, i.e. the equality \(C_{[\alpha ]}=C_{[\beta ]}\)  does not in general imply that cosets \(\alpha \)  and \(\beta \)  belong to the same coadjoint orbit.
Example 4.7 Consider the disk with two holes from Figure REF (a). In this case there are no circulation functions since there are no solid edges in the Reeb graph. On the other hand, in this case there are no nontrivial symplectomorphisms preserving the function hence the dimension of the space of coadjoint orbits is equal to the first Betti number of the surface, i.e. it is equal to two.

It turns out that it is possible to define some additional invariants: integrals of cosets over certain cycles associated with the pair \((M,F)\)  in an invariant way.
There is a unique way to lift each edge \(e\subset \Gamma _F^d\)  to a smooth oriented (and diffeomorphic to a segment) curve \(\tilde{e}\subset \partial M\)  such that
\(\pi (\tilde{e})=e;\)

for each \(x\in e\setminus \partial e\)  the regular \(F\) -level \(\pi ^{-1}(x)\)  is pointed in the direction of the curve \(\tilde{e}.\) 

We define the subset \(\tilde{E}_F\subset M\)  to be the union
\(\tilde{E}_F:=\bigcup _{e\in E(\Gamma ^d_F)}\tilde{e}.\) 
We also define the subset \(\tilde{V}_F\subset M\)  to be
\(\tilde{V}_F:=\pi ^{-1}(V(\Gamma ^d_F)\setminus \partial \Gamma ^d_F)\) 
where \(\partial \Gamma ^d_F\)  is the set of boundary vertices (i.e. vertices of types I, III, or V) of the graph \(\Gamma ^d_F.\)  And, finally, define the subset \(\tilde{\Gamma }_F\)  to be the union of \(\tilde{E}_F\)  and \(\tilde{V}_F\)  (see Figure REF ).
<FIGURE>The set \(\tilde{\Gamma }_F\)  is a topological graph embedded into the surface \(M.\)  We denote by \(i\)  the inclusion
\(\tilde{\Gamma }_F{} M.\)
Lemma 4.8 
The map \(\pi \circ i\colon \tilde{\Gamma }_F\rightarrow \Gamma _F^d\)  is a homotopy equivalence.

Consider the graph \(\tilde{\Gamma }_F/\tilde{V}_F\)  obtained from \(\tilde{\Gamma }_F\)  by contracting each connected component of a singular \(F\) -level in \(\tilde{\Gamma }_F\)  to a point. Denote by \(p\)  the projection \(\tilde{\Gamma }_F\rightarrow \tilde{\Gamma }_F/\tilde{V}_F.\)  The map \(p\)  is a homotopy equivalence since each singular \(F\) -level in \(\tilde{\Gamma }_F\)  is connected and simply connected. The map
\(i\circ \pi \)  factors (in a unique way) through \(\tilde{\Gamma }_F/\tilde{V}_F,\)  i.e. there exists a unique map \(\tilde{\pi }:\tilde{\Gamma }_F/\tilde{V}_F\rightarrow \Gamma _F^d\)  such that
\(\pi \circ i=\tilde{\pi }\circ p \circ i.\) 
The map \(\tilde{\pi }\)  is a homeomorphism. We conclude that the map \(p\circ i\)  is a homotopy equivalence as a composition of the inclusion \(i,\)  the homotopy equivalence \(p,\)  and the homeomorphism \(\tilde{\pi }.\) 

Let \([\alpha ]\in \Omega ^1(M)/\mathrm {d} \Omega ^0(M)\)  be a coset of a one-form.
There is a natural way to define the restriction \(i^*[\alpha ]\in \mathrm {H}^1(\tilde{\Gamma }_F).\) 
First, we define the restriction \(i^*\alpha \)  as a one-cochain such that
\(i^*\alpha (e):=\int _{e}\alpha \) 
for each edge \(e\subset \tilde{\Gamma }_F.\) 
Now we take \(i^*[\alpha ]:=[i^*\alpha ].\)  The cohomology class \(i^*[\alpha ]\)  is well-defined since each exact one-form \(\mathrm {d} f\)  restricts to the exact one-cochain \(i^*\mathrm {d} f.\)  It follows from Lemma REF  that
\(i^*\circ \pi ^*\colon \mathrm {H}^1(\Gamma _F)\rightarrow \mathrm {H}^1(\tilde{\Gamma }_F)\) 
is an isomorphism. Hence with each coset
\([\alpha ]\in \Omega ^1(M)/\mathrm {d} \Omega ^0(M)\) 
we can also associate an element \(\xi _{[\alpha ]}\in \mathrm {H}^1(\Gamma _F)\) 
defined by the formula
\(\xi _{[\alpha ]}:=(i^*\circ \pi ^*)^{-1}(i^*[\alpha ]).\) 
Next, we generalize the notion of a circulation graph from [5]}.
Definition 4.9 A measured Reeb graph \((\Gamma , f,\mu )\)  endowed with a circulation function \(\pazocal {C}\)  and an element
\(\xi \in \mathrm {H}^1(\Gamma ^d)\) 
is called a augmented circulation graph
\((\Gamma ,f,\mu ,  \pazocal C,\xi ).\)

We demonstrated above that with each coset \([\alpha ]\)  one can associate an augmented circulation graph \(\Gamma _{[\alpha ]}.\) 
Two augmented circulation graphs are isomorphic if they
are isomorphic as measured Reeb graphs, and the isomorphism between them preserves all additional data. An augmented circulation graph
\((\Gamma ,f,\mu ,  \pazocal C,\xi )\) 
is compatible with a symplectic surface \((M,\omega )\)  if the corresponding measured Reeb graph \((\Gamma ,f,\mu )\)  is compatible with \((M,\omega )\)  (see Definition REF ).
Coadjoint orbits of symplectomorphism groups
Theorem 4.10 
Let \((M,\omega )\)  be a connected symplectic surface with or without boundary. Then generic coadjoint orbits of \(\mathrm {SDiff}(M)\)  are in one-to-one correspondence with (isomorphism classes of) augmented circulation graphs
\((\Gamma , f, \mu ,  \pazocal C)\)  compatible with \(M\) .
In other words, the following statements hold:
For a symplectic surface \(M\)  and generic cosets \([\alpha ], [\beta ] \in \mathfrak {s}\mathfrak {vect}^*(M)\) 
the following conditions are equivalent:

\([\alpha ]\)  and \( [\beta ]\)  lie in the same orbit of the \(\mathrm {SDiff}(M)\)  coadjoint action;

augmented circulation graphs \(\Gamma _{[\alpha ]}\)  and \(\Gamma _{[\beta ]}\)  corresponding to the cosets \([\alpha ]\)  and \([\beta ]\) 
are isomorphic.

For each augmented circulation graph \(\Gamma \)  which is compatible with \(M\) , there exists a generic \([\alpha ] \in \mathfrak {s}\mathfrak {vect}^*(M)\)  such that \(\Gamma _{[\alpha ]} =(\Gamma , f, \mu ,  \pazocal C, \xi )\) .
Corollary 4.11 The space of coadjoint orbits of the group \(\mathrm {SDiff}(M)\)  corresponding to the same measured Reeb graph \((\Gamma ,f,\mu )\)  is a finite-dimensional affine space and its dimension is \(\dim \mathrm {H}_1(\Gamma ,\Gamma ^d) + \dim \mathrm {H}_1(\Gamma ^d).\)

Remark 4.12 It follows from the long exact sequence for the pair \((\Gamma ,\Gamma ^d)\)  that
\(\dim \mathrm {H}_1(\Gamma ,\Gamma ^d) + \dim \mathrm {H}_1(\Gamma ^d)=\dim \mathrm {H}_1(\Gamma ) - \dim \mathrm {H}_0(\Gamma ^d)+1.\) 
Therefore, the space of coadjoint orbits of the group \(\mathrm {SDiff}(M)\)  corresponding to the same measured Reeb graph \((\Gamma ,f,\mu )\)  has dimension \(\dim \mathrm {H}_1(\Gamma )\)  in the case when the subgraph \(\Gamma ^d\)  is connected.

Example 4.13 
Consider the torus with one boundary component from Figure REF  with the height function \(F\)  on it, and the corresponding Reeb graph \(\Gamma _F\) . In this case \(\mathrm {H}_1(\Gamma _F^d)=0\)  and \(\mathrm {H}_1(\Gamma _F,\Gamma _F^d)=1.\)  Therefore, the corresponding space of coadjoint orbits is one-dimensional.

Before we proceed with the proof of Theorem REF  let us formulate and prove two lemmas.
Lemma 4.14 
Let \(M\)  be a connected oriented surface with non-empty boundary, and let \(F\)  be a simple Morse function on \(M\) . Then
\(\dim \mathrm {H}_1(M^d_F) = \dim \mathrm {H}_1(\Gamma ^d_F)+ \dim \mathrm {H}_0(\Gamma ^s_F\cap \Gamma ^d_F).\) 

Let \(\tilde{M}\)  be the smooth surface obtained from the surface \(M^d_F\)  by
contracting each circle in \(M^d_F\cap M^s_F\)  to a point. It is clear that
\(\dim \mathrm {H}_1(M^d_F)=\dim \mathrm {H}_1(\tilde{M})+ \dim \mathrm {H}_0(\Gamma ^s_F\cap \Gamma ^d_F).\) 
Let \(p\)  be the canonical projection \(M\rightarrow \tilde{M}.\)  The function \(F\)  descends to a simple Morse function \(\tilde{F}\colon \tilde{M}\rightarrow {\mathbb {R}}\)  such that
\(F=\tilde{F}\circ p.\) 
The Reeb graph \(\Gamma _{\tilde{F}}\)  consists only of dashed edges, and it is coincides with \(\Gamma ^d_F.\)  Then the surface \(\tilde{M}\)  is homotopy equivalent to graph \(\Gamma _{\tilde{F}}.\)  Therefore,
\(\begin{aligned}& \dim \mathrm {H}_1(M^d_F)=\dim \mathrm {H}_1(\tilde{M})+ \dim \mathrm {H}_0(\Gamma ^s_F\cap \Gamma ^d_F)= \dim \mathrm {H}_1(\Gamma _{\tilde{F}})+ \dim \mathrm {H}_0(\Gamma ^s_F\cap \Gamma ^d_F)\\& =\, \dim \mathrm {H}_1(\Gamma ^d_F)+ \dim \mathrm {H}_0(\Gamma ^s_F\cap \Gamma ^d_F).\end{aligned}\) 
Lemma 4.15 
Let \(M\)  be a connected oriented surface possibly with boundary, and let \(F\)  be a simple Morse function on \(M\) .
Assume that \([\gamma ] \in \mathrm {H}^1(M)\)  is such that the integral of \(\gamma \)  over
any \(F\) -level vanishes, and \(\xi _{[\gamma ]}\)  is a zero element in \(\mathrm {H}^1(\Gamma _F^d).\) 
Then there exists a \(C^\infty \)  function \(H \colon M \rightarrow {\mathbb {R}}\)  (with zero restriction on the surface \(M^d_F\) ) such that the one-form \(H\mathrm {d} F\)  is closed, and its cohomology class
is equal to \([\gamma ]\) . Moreover, \(H\)  can be chosen in such a way that the ratio
\(H/F\)  is a smooth function.

Denote by \(i_d\)  the inclusion \(M_F^d\cap M_F^s {} M^d_F,\)  and denote by \(\pi _d\)  the restriction of the projection \(\pi \colon M\rightarrow \Gamma _F\)  on the surface \(M^d_F.\)  Note that the homomorphism
\((\pi _d)_*\colon \mathrm {H}_1(M_F^d) \rightarrow \mathrm {H}_1(\Gamma _F^d)\) 
is a surjection, and
\(\mathrm {Im} \, (i_d)_* \subset \mathrm {Ker} \, (\pi _d)_*.\) 
It follows from Lemma REF  that
\(\dim \mathrm {H}_1(M^d_F) = \dim \mathrm {H}_1(\Gamma ^d_F)+ \dim \mathrm {H}_0(\Gamma ^s_F\cap \Gamma ^d_F).\) 
Hence the image of the homomorphism \((\pi _d)_*\)  coincides with the kernel of the homomorphism \((\pi _d)_*.\) 
From above we conclude that the homomorphism
\(\pi _*\colon \mathrm {H}^1(\Gamma _F^d) \rightarrow \mathrm {H}^1(M_F^d)\) 
is an injection, and
\(\mathrm {Im} \, (\pi _d)^* = \mathrm {Ker} \, (i_d)^*.\)
Denote by \(i\)  the inclusion \(M_F^d {} M_F.\) 
Since the integral of \([\gamma ]\)  over any connected component of any closed \(F\) -level vanishes
and \(\xi _{[\gamma ]}\)  is a zero element in \(\mathrm {H}^1(\Gamma _F^d),\) 
the cohomology class \([i^*\gamma ]\)  is a zero element in \(\mathrm {H}^1(M_F^d).\) 
Consider the long exact cohomology sequence for the pair \((M_F, M^d_F)\colon \)
\(0 &\rightarrow \mathrm {H}^{0}(M_F^d) &\rightarrow \mathrm {H}^{0}(M_F)\rightarrow \mathrm {H}^1(M_F,M^d_F) \rightarrow \mathrm {H}^1(M_F) &\rightarrow \mathrm {H}^1(M^d_F) \rightarrow 0.\) 

The cohomology class \([\gamma ]\)  on \(M\)  belongs to the kernel of the homomorphism
\(i^*\colon \mathrm {H}^1(M_F)\rightarrow \mathrm {H}^1(M^d_F).\) 
Hence it belongs to the image of the homomorphism
\(\mathrm {H}^1(M_F,M^d_F) \rightarrow \mathrm {H}^1(M_F),\) 
i.e. there exists a one-form \(\tilde{\gamma }\)  such that
\([\tilde{\gamma }]=[\gamma ]\)  and \(\tilde{\gamma }|_{M^d_F}=0.\)
Denote by \(\pi _s\)  the restriction of the projection \(\pi \colon M\rightarrow \Gamma _F\)  on the surface \(M^s_F.\)  The homomorphism
\((\pi _s)_*\colon \mathrm {H}_1(M_F^s) \rightarrow \mathrm {H}_1(\Gamma _F^s)\) 
is a surjection, and its kernel is generated by those homology classes which are homologous to regular \(F\) -levels. From above we conclude that the homomorphism
\((\pi _s)^*\colon \mathrm {H}^1(\Gamma _F^s) \rightarrow \mathrm {H}^1(M_F^s)\) 
is an injection, and
\(\mathrm {Im} \, (\pi _s)^*=\mathrm {Ann} \, \mathrm {Ker} \, (\pi _s)_*\) 
where
\(\mathrm {Ann} \, \mathrm {Ker} \, (\pi _s) := \left\lbrace  \omega \in \mathrm {H}^1(\Gamma _F^s) \mid \omega (c)=0 \iff c \in \mathrm {Ker} \, (\pi _s)_* \right\rbrace .\) 
Therefore, there exists a one-cochain
\(\alpha \in \mathrm {H}^1(\Gamma ^s_F)\) 
of the form
\(\alpha =\sum _{e\in E(\Gamma ^s_F)}\alpha _e e^*\) 
such that \([\tilde{\gamma }]=(\pi _s)^*[\alpha ].\)
Recall that the function \(f\)  is the pushforward of the function \(F\)  to the graph \(\Gamma _F.\)  Consider a continuous function \( h \colon \Gamma _F \rightarrow {\mathbb {R}}\)  such that

it is a smooth function of \(f\)  in a neighborhood of each point \(x \in \Gamma _F\) ;

it vanishes whenever \(x\)  is sufficiently close to a vertex;

\(h|_{\Gamma ^d_F}=0;\)

for each edge \(e\) , we have
\(\alpha (e) = \int _e \! h\mathrm {d} f.\) 

Obviously, such a function does exist. Now, lifting \( h\)  to \(M\) , we obtain a smooth function \(H\)  with
the desired properties.

[Proof of Theorem REF ]
Let us prove the first statement. The implication (a) \(\Rightarrow \)  (b) is immediate,
so it suffices to prove the implication (b) \(\Rightarrow \)  (a).
Let \(\phi \colon \Gamma _{[\alpha ]} \rightarrow \Gamma _{[\beta ]}\)  be an isomorphism
of augmented circulation graphs. By Theorem REF , \(\phi \)  can be lifted to a symplectomorphism
\( \Phi \colon M \rightarrow M\)  that maps the function \(F = \mathfrak {curl}[\alpha ]\) 
to the function \(G = \mathfrak {curl}[\beta ]\) . Therefore, the 1-form \(\gamma \)  defined by
\(\gamma = \Phi ^*\beta - \alpha \) 
is closed.
Assume that \( \Psi \colon M \rightarrow M \)  is a symplectomorphism which maps the function \(F\)  to itself and is isotopic to the identity. Then the composition \(\widetilde{\Phi }= \Phi \circ \Psi ^{-1}\)  maps \(F\)  to \(G\) , and
\([\widetilde{\Phi }^*\beta - \alpha ] = [\Phi ^*\beta - \Psi ^*\alpha ] = [\gamma ] - [\Psi ^*\alpha - \alpha ].\) 
We claim that \(\Psi \)  can be chosen in such a way that \(\widetilde{\Phi }^*\beta - \alpha \)  is exact, i.e. one has the equality of the cohomology classes
\([\Psi ^*\alpha - \alpha ] = [\gamma ].\) 
Moreover, we show that there exists a time-independent symplectic vector field \(X\)  that
preserves \(F\)  and satisfies
\([\Psi _t^*\alpha - \alpha ] = t[\gamma ]\,,\) 

where \(\Psi _t\)  is the flow of \(X\) .
Differentiating (REF ) with respect to \(t\) , we get in the left-hand side
\([\Psi _t^*L_X\alpha ] = [L_X\alpha ] = [i_X\mathrm {d} \alpha ] = [F\cdot i_X \omega ]\,,\) 
since the form \(L_X \alpha \)  is closed and \(\Psi _t^*\)  does not change its cohomology class. Thus
\([F\cdot i_X \omega ] = [\gamma ].\) 

Since \(\Phi \)  preserves the circulation function, the integrals of \(\gamma \)  over all connected components of \(F\) -levels vanish.
In addition, \(\xi _{[\Phi ^*\alpha ]}=\xi _{[\alpha ]}.\) 
Therefore, by Lemma REF , there exists a smooth function \(H\)  such that
\([\gamma ] = [H\mathrm {d} F].\) 
Now we set
\(X := \frac{H}{F} \,\omega ^{-1}\mathrm {d} F.\) 
It is easy to see that the vector field \(X\)  is zero on \(M^d,\)  symplectic, preserves the levels of \(F\) , and satisfies the equation (REF ). Therefore, its phase flow map \(\Psi =\Psi _1\)  has the required properties.
Now let us prove the second statement. It follows from Theorem REF  that there exists a symplectic surface \((M,\omega )\)  and a simple Morse function \(F\colon M\rightarrow {\mathbb {R}}\)  such that \(\Gamma _F=\Gamma .\)  Consider the surface \(M^s\)  and the restriction \(F|_{M^s}\)  of the function \(F\)  to the surface \(M^s.\)  The restriction \(F|_{M^s_F}\)  is a Morse function, and it is constant on the boundary \(\partial M^s_F\)  since it is formed by some of closed \(F\) -levels. However, it is not necessarily a function of \(CB\) -type since it has hyperbolic critical points on the boundary whenever the graph \(\Gamma _F\)  has vertices of type \(V.\)  In order to apply the Theorem REF 
we need to `cut out' from \(M^s\)  these hyperbolic critical points. Let \(v\in \Gamma _F\)  be a vertex of type \(V,\)  let \(e\subset \Gamma _F\)  be the only solid edge incident to \(v,\)  and also let \(u\in \Gamma ^s_F\)  be the only other vertex adjacent to \(e.\)  The edge \(e\) , with endpoints \(\lbrace v,u\rbrace \)  can be uniquely subdivided into two edges, say \(e_{v\rightarrow w}\)  and \(e_{w\rightarrow u}\) , connecting to a new vertex \(w\)  such that \(\mu (e_{v\rightarrow w})=\mu (e_{w\rightarrow u}).\)  After that we cut out the edge \(e_{v\rightarrow w}.\)  Denote by \(\Gamma ^{\prime }\)  the (abstract) measured Reeb graph obtained by applying the above procedure to all vertices of type \(V\)  in the graph \(\Gamma _F\)  (see Figure REF ).
<FIGURE>Denote by \(M^{\prime }\subset M^s_F\)  the preimage
\(\pi ^{-1}(\Gamma ^{\prime })\) 
It is clear from the above that the restriction
\(F|_{\tilde{M}}\)  is a function of \(CB\) -type. Therefore, it follows from Theorem REF  that there exists a one-form \(\alpha _0\)  on \(M^{\prime }\)  such that
\( \pazocal C_{[\alpha _0]}= \pazocal C|_{\Gamma ^{\prime }}.\) 
It is clear that the form \(\alpha _0\)  can be extended (along the cylinders \(M^s\setminus M^{\prime }\) ) on all of \(M^s\)  in such a way that
\( \pazocal C_{[\alpha _0]}= \pazocal C|_{\Gamma }.\) 
On the other hand, there exists a one-form \(\alpha _1\)  on \(M\)  such that
\([(i_d)^*\alpha _1]=[(i_s)^*\alpha _0]\) 
and
\(\xi _{[\alpha ]}=\xi \) 
since
\(\dim \mathrm {H}_1(M^d_F) = \dim \mathrm {H}_1(\Gamma ^d_F)+ \dim \mathrm {H}_1(M^d_F\cap M^s_F).\) 
Using an appropriate partition of unity we construct a one-form \(\alpha \)  (as a combination of one-forms
\(\alpha _0\)  and \(\alpha _1\) ) such that
\( \pazocal C_{[\alpha ]}= \pazocal C_{[\alpha _0]}= \pazocal C\) 
and
\(\xi _{[\alpha ]}=\xi _{[\alpha _1]}=\xi .\) 
Hence the augmented circulation graph \(\Gamma _{[\alpha ]}\)  coincides with \(\Gamma .\)
Conclusion
In this paper we have classified simple Morse functions on symplectic surfaces and generic coadjoint orbits of symplectomorphism groups of surfaces.
This allowed us to completely resolve the questions posed in [6]}.
The answer to Problem \(5.5\)  on reconstruction of a surface with boundary from its Reeb graph was given in the work [7]}. As described in Section  the idea is to add a cyclic order for the dashed edges incident to II- or IV-vertices to the structure of an abstract Reeb graph in order to reconstruct the corresponding surface. Table REF  gives an answer to Problem \(5.6\)  on measure asymptotics on the graph. The required by Problem \(5.7\)  compatibility conditions are described in Definition REF . And finally, Theorem REF  describes the required by Problem \(5.8\)  additional invariants for coadjoint orbits in case of surfaces with boundary.
One should mention two other relevant but not overlapping with us classification results for symplectic surfaces:

Dufour, Molino, and Toulet classified in [8]} simple Morse fibrations on closed symplectic surfaces under the action of symplectic diffeomorphisms.

Bolsinov [9]} and Kruglikov [10]} classified Hamiltonian vector fields tangent to the boundary on surfaces up to the action of arbitrary diffeomorphisms.

It would be interesting to extend those classifications for surfaces with boundary. (Note that in [10]} in the contrast with the present work Hamiltonian functions are assumed to be constant on the boundary.)
It also would be very interesting to classify Morse functions and Morse orbits for the action of

the group \(\mathrm {Ham}(M)\)  of Hamiltonian diffeomorphisms of a surface \(M;\)

the connected component \(\mathrm {SDiff}_0(M)\)  of the identity in the group \(\mathrm {SDiff}(M)\)  for the case of surfaces \(M\)  with boundary.

This would generalise the corresponding results of [5]} and the present work to these important subgroups of the symplectomorphism groups.

Euler's equation and coadjoint orbits
In this Appendix we describe following [5]} the hydrodynamical motivation of the above classification problems. Consider a symplectic surface \((M,\omega )\)  with boundary \(\partial M.\)  We denote by \(\mathrm {SDiff}(M)\)  the Lie group of all symplectomorphisms of \(M,\)  and by \(\mathfrak {s}\mathfrak {vect}(M)\)  the corresponding Lie algebra of divergence-free vector fields on \(M.\)  A linear functional \(I\)  on \(\mathfrak {s}\mathfrak {vect}(M)\)  is called regular if there exists a smooth 1-form \(\xi _I\)  such that the value of \(I\)  on a vector field \(v\)  is the pairing between \(\xi _I\)  and \(v:\)
\(I(v)=\int _M \xi _I(v)\omega .\) 
The space \(\mathfrak {s}\mathfrak {vect}^*(M)\)  of regular functionals on \(\mathfrak {s}\mathfrak {vect}(M)\)  is a dense subset in the space of all continues linear functionals on \(\mathfrak {s}\mathfrak {vect}(M).\)  It turns out that the space of regular functionals \(\mathfrak {s}\mathfrak {vect}^*(M)\)  can be identified with the space of cosets
\(\Omega ^1(M)/d\Omega ^0(M),\)  since exact 1-forms give zero functionals on divergence-free vector fields. Moreover, the natural action of the group \(\mathrm {SDiff}(M)\)  on the space of cosets \(\Omega ^1(M)/d\Omega ^0(M)\)  by means of pull-backs coincides with the coadjoint action of the group of symplectomorphisms \(\mathrm {SDiff}(M)\) . The proof of this fact can be found in [14]} (see Section I.8). More information about infinite-dimensional Lie groups can be found in [15]}.
Now let us fix a Riemannian metric \((\cdot ,\cdot )\)  on the surface \(M\)  such that the corresponding area form coincides with the symplectic form \(\omega .\)  The motion of an inviscid incompressible fluid on \(M\)  is described by the Euler equation
\(\partial _t v+\nabla _v v = -\nabla p\) 
describing an evolution of a divergence-free velocity field \(v\)  of a fluid flow in \(M\) , where
\({\rm div}\, v=0\)  and the field \(v\)  is tangent to the boundary \(\partial M.\) 
The pressure function \(p\)  entering the Euler equation is defined uniquely modulo
an additive constant by this equation along with the divergence-free constraint
on the velocity \(v\) .
The metric \((\cdot ,\cdot )\)  allows us to
identify the (smooth parts of) the Lie algebra and its dual by means of the so-called inertia operator:
given a vector field \(v\)  on \(M\)  one defines the 1-form \(\alpha =v^\flat \) 
as the pointwise inner product with vectors of the velocity field \(v\) :
\(v^\flat (W) := (v,W)\)  for all \(W\in T_x M.\) 
The Euler equation (REF ) rewritten on 1-forms is
\(\partial _t \alpha +L_v \alpha =-dP\,\) 
for the 1-form \(\alpha =v^\flat \)  and an appropriate function \(P\)  on \(M\) .
In terms of the cosets of 1-forms \([\alpha ]=\lbrace \alpha +df\,|\,f\in C^\infty (M)\rbrace \in \Omega ^1(M) / \mathrm {d} \Omega ^0(M)\) , the Euler equation looks as follows:
\(\partial _t [\alpha ]+L_v [\alpha ]=0\) 
on the dual space \(\mathfrak {g}^*\) , where \(L_v\)  is the Lie derivative along the field \(v\) .
The Euler equation (REF ) shows that the coset of 1-forms \([\alpha ]\)  evolves by an area-preserving change of coordinates, i.e. it remains in the same coadjoint orbit in \(\mathfrak {g}^*\) . This is why
invariants of coadjoint orbits of cosets \([\alpha ]\)  describe first integrals, called Casimirs, of the Euler equation, and their complete classification is important in many areas of ideal fluid dynamics.


tocsectionReferences
